The present invention relates generally to the analysis of the steady-state behavior of circuits driven by a periodic input signal.
The rapid growth in the field of wireless RF telecommunications has provided significant motivation for the development of simulation tools that can quickly and accurately analyze both the steady-state behavior and the response to multiple-frequency excitations of nonlinear circuits. FIG. 1A depicts a generic circuit 10 that is frequently analyzed in engineering applications. Circuit 10 is driven by a periodic input signal 11, u(t), of arbitrary shape, and has an output 12, v(t), that is periodic with the same frequency as u(t) when circuit 10 has reached a steady state. There is a great need for computationally efficient techniques for determining the steady-state behavior of a circuit such as circuit 10, particularly given the fact that frequently the circuit behavior must be characterized over a wide range of input signal frequencies and amplitudes.
One traditional technique for determining a periodic steady-state solution (PSS) is transient analysis. This technique involves determining respective states of the circuit at a series of timepoints (starting at time zero with a set of initial conditions), the state determined for the circuit at any particular timepoint depending on the state determined for the circuit at the previous timepoint. The technique terminates upon detecting that a steady-state has been reached. Transient analysis requires several timepoints per period of the input signal, and thus is often impractical in circuits having a time constant many orders of magnitude greater than the period of the input signal. For example, FIG. 1B depicts a self-biased amplifier circuit 20 that is driven by a periodic input signal 21, u(t). The time constant of circuit 20 might be chosen to be several orders of magnitude larger than the largest possible period for u(t), thereby requiring transient analysis over thousands of timepoints.
Another prior art technique for obtaining the PSS solution for a circuit involves the use of the shooting-Newton method. This method uses a Newton iterative technique to compute a series of estimates for the PSS. The difference between each estimate in the series and the previous estimate is determined by solving a respective linear system of equations via Gaussian elimination. Obtaining the solution for each linear system by Gaussian elimination involves factorization of a dense N by N matrix, where N represents the number of nodes in the circuit, and thus requires a computation time that increases with the third power of N. For this reason, prior art shooting-Newton techniques for obtaining a PSS solution are often impractical for circuits with a large number of nodes.
Another prior art technique for obtaining a PSS solution involves the use of a finite-difference method. Such a method discretizes the circuit differential equations at a set of timepoints spanning an interval of the input signal to obtain a set of difference equations. Solution of this set of difference equations provides a PSS solution for each of the timepoints simultaneously. Like the previous PSS methods discussed, PSS techniques based on finite-difference methods typically have a computational cost that grows as the cube of the number of circuit nodes, and thus also may be impractical for large circuits.
FIG. 1C depicts another type of generic circuit 30 that is frequently analyzed in engineering applications. Circuit 30 is driven by two periodic input signals 31-32, u.sub.l and u.sub.s. u.sub.s is a small sinusoidal signal of much smaller amplitude than u.sub.l. u.sub.l is periodic with arbitrary shape. One approach to the solution of circuit 30 is transient analysis. Transient analysis may be a very expensive option, particularly where one of the input signals is of much higher frequency than the other. For example, FIG. 1D depicts a switched-capacitor filter circuit 40 driven by a periodic signal u.sub.s to be filtered and a high frequency clock signal u.sub.l that controls the state of a transistor 41. To ensure accurate results, transient analysis must cover several timepoints per period of the high frequency clock u.sub.l, even though the frequency of the filtered signal u.sub.s is much lower.
Another approach to studying the behavior of circuit 30, known as Periodic Time-Variant Analysis (PTV), treats the small input signal, u.sub.s, as a small perturbation to the periodic steady-state (PSS) response of circuit 30 to u.sub.l when u.sub.s =0. In particular, typically, a small-signal steady-state response is determined at a set of time points spanning a period of u.sub.l, by linearizing the differential equations for circuit 30 about the PSS response, and then discretizing the resulting system at the set of time points. Such a technique has a computational cost that grows with the third power of the number of circuit nodes, and for this reason, is impractical for large circuits.
FIG. 1E depicts another generic circuit 50 of a type frequently analyzed in engineering applications (e.g., mixer circuits). Circuit 50 is driven by d periodic signals 51 of unrelated frequencies, and typically has a steady-state response 52 that is quasiperiodic in d frequencies. One approach to determining a quasiperiodic response involves the use of mixed frequency-time methods as described in detail in chapter 7 of the text "Steady-State Methods for Simulating Analog and Microwave Circuits" by Kundert et al. However, such methods typically have a computational cost that grows with the third power of the number of circuit nodes, and for this reason, are impractical for large circuits.